Intersection homology II (original) (raw)
Related papers
Refined intersection homology on non-Witt spaces
Journal of Topology and Analysis, 2015
We investigate a generalization to non-Witt stratified spaces of the intersection homology theory of Goresky–MacPherson. The second-named author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we first extend both of these cohomology theories by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. Our main result is that this refined intersection cohomology theory coincides with the analytic de Rham theory on Thom–Mather stratified spaces. The word "refined" is motivated by the fact that the definition of this cohomology theory depends on the choice of an additional structure (mezzo-perversity) which is automatically zero in the case of a Witt space.
Homology stratifications and intersection homology
Proceedings of the Kirbyfest, 1999
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersection homology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky-MacPherson proof valid for PL spaces, and we ask a number of questions. The proof uses a new technique, homology general position, which sheds light on the (open) problem of defining generalised intersection homology.
Intersection Cohomology on Nonrational Polytopes⋆
Compositio Mathematica, 2003
Abstract. We consider a fan as a ringed space (with finitely many points). We develop the cor-responding sheaf theory and functors, such as direct image Rp√ (p is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf LF, called the minimal sheaf plays the ...
The hypertoric intersection cohomology ring
Inventiones Mathematicae, 2009
We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
Intersection cohomology of pure sheaf spaces using Kirwan’s desingularization
Journal of Geometry and Physics
Let Mn be the Simpson compactification of twisted ideal sheaves IL,Q(1) where Q is a rank 4 quardric hypersurface in P n and L is a linear subspace of dimension n − 2. This paper calculates the intersection Poincaré polynomial of Mn using Kirwan's desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree ≥ 8 by considering wall-crossings of stable pairs and complexes.
2006
The contents of these notes originated from a talk delivered to a diversified audience at the Departamento de Matemática of my own University. The present notes expand and give full details of statements made in that occasion. Moreover, I have sought to expand as well on several algebraic notions in order to pave the way towards a more stable version even though self-sufficiency is out of question in a material of this order. If one is to single a module that stands alone in its totally device-independent nature, one'd better take the module of differentials Ω(A/k). In fact, most invariants of an algebra or scheme are attached primevally to this module or "découlent" from it by various procedures. Thus, Segre classes in algebraic geometry are ultimately defined in terms of Ω(A/k) and even the ubiquitous canonical module (dualizing sheaf) is "normally" related to Ω(A/k) by double-dualizing its top-wedge module. My objective in this Course is panorama-driven, due to the short span of lectures. However, details and examples will be worked out and, possibly, a few arguments too. 10. * Considerations about a formula of Kleiman-Plücker-Teissier for the degree (class) of the dual variety to a complete intersection.
Vanishing Theorems for Constructίble Sheaves Π
2005
In [H-L2], we have proved theorems on the vanishing of the higher cohomology of direct images of complexes with constructible cohomology sheaves (constructible complexes). As a particular case of these theorems, we have obtained a theorem of Artin-Grothendieck which states that the direct image by an affine map of a constructible sheaf satisfying the support condition ([B-B-D] (4.0.1) or [G-M2] §4.1 Definition), also satisfies the support condition. P. Deligne has obtained the Lefschetz Theorem for hyperplane sections for the Z/cohomology (cf [D2] (4.1.6)) by considering the theorem of Artin-Grothendieck in the case the affine map has a point as target and by using the duality. In fact, Lefschetz Theorem can be obtained directly by proving theorems on the vanishing of the lower cohomology of direct images of constructible complexes. Following Deligne's idea, one would use duality theorems and apply them to the results of [H-L2], as K. Fieseler and L. Kaup do in [F-K2] for the in...